4.6 Exercises

20 points · 4 extra credit

Rank

1. 1. (3 pts) For the matrix $A = \begin{bmatrix} 1 & 2 & 1 & 1 & 0 \\ 2 & 4 & 3 & 0 & 1 \\ 1 & 2 & 2 & 0 & 3 \end{bmatrix},$ row-reduce $A$ to echelon form and find $\operatorname{rank} A$ and $\dim\operatorname{Nul} A$. Identify the pivot columns and the free columns, give a basis for $\operatorname{Col} A$, and confirm that $\operatorname{rank} A + \dim\operatorname{Nul} A$ equals the number of columns.

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2. 2. (2 pts) Let $A$ be a $5\times 9$ matrix with $\dim\operatorname{Nul} A = 4$. Find $\operatorname{rank} A$.

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3. 3. (2 pts) Let $A$ be a $7\times 5$ matrix with $\operatorname{rank} A = 5$. Find $\dim\operatorname{Nul} A$, and state whether the columns of $A$ are linearly independent.

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4. 4. (3 pts) Let $A$ be a $4\times 6$ matrix with $\operatorname{rank} A = 3$. Find $\dim\operatorname{Nul} A$, $\dim\operatorname{Row} A$, $\dim\operatorname{Col} A$, and $\dim\operatorname{Nul} A^{\mathsf T}$.

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5. 5. (2 pts) Can a $4\times 7$ matrix have rank $5$? Explain.

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6. 6. (3 pts) State whether each of the following is true or false, and justify each answer.

   (a) $\operatorname{rank} A$ equals the number of pivot columns of $A$.

   (b) For an $m\times n$ matrix, $\operatorname{rank} A + \dim\operatorname{Nul} A = n$.

   (c) The row space and column space of a matrix always have the same dimension.

   (d) For an $m\times n$ matrix, $\operatorname{rank} A \le \min(m, n)$.

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7. 7. (3 pts) Explain, using the reduced echelon form of an $m\times n$ matrix $A$, why $\dim\operatorname{Col} A$ equals the number of pivot columns and $\dim\operatorname{Nul} A$ equals the number of free variables, and hence why $\operatorname{rank} A + \dim\operatorname{Nul} A = n$.

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8. 8. (2 pts) Let $A$ be a $3\times 5$ matrix. Explain why the system $A\mathbf{x} = \mathbf{b}$ can never have a unique solution (assuming it is consistent).

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9. 9. (4 pts ••) Prove that $\operatorname{rank} A = \operatorname{rank} A^{\mathsf T}$ for every matrix $A$ (equivalently, row rank equals column rank).

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